From what I've gathered, the public's general negative view towards furries, and the idea that we're "freaks" or that we're "gross" is mainly because of misconceptions. For some reason, people seem to think that furry fandom is exclusively like a sex fetish, like we're "animal humpers," (i.e. zoophiliacs), or that we dress up in fursuits strictly as a turn on for our partners.
In fact, if you want an idea of how bad these misconceptions are, Google the CSI episode that takes place at a furry convention. It's like people think that fursuiters all get together to have one giant orgy, which is completely false. And while some furries might be zoophiliacs (I personally am not), from what I understand zoophiles don't abuse animals or force them into unwanted acts. And while fursuiting might be a turn on for a lot of us, I'm sure we can all agree that it's not the main purpose of furry fandom.
I'd say that you can safely call yourself a "furry" ggoldduck. Furry fandom has a wide spectrum, from people who simply have an interest in anthro characters and art, to people who feel beyond a doubt, like they are an animal living in a human body. I don't think there's anything wrong with with being anywhere on this spectrum. We're just different, but that also makes us special.
And we're all contributing members to society, and we don't hurt anyone, so there is nothing wrong with being who we are. And I love connecting with people who share this interest.
Me personally, I love anthro art, and can relate a lot to the canine species (like you were talking about), so I consider my fursona as sort of an alter-ego. My girlfriend as also a furry, and I think she feels the same way. And as for why a lot of us wear fursuits...well, some people are gung-ho into things like Star Trek, and go to conventions and dress up like the characters, or even wear that stuff in public. So how are we any different?
And as for "yiffing" and the fetish side of furries, well...I guess if there's a deep interest in these characters, to various degrees, there's also an attraction to them, which there is also nothing wrong with.
Anyway to answer your question, I'm not very open about the subject, except to communities like this online. I don't see the point in sharing this with people if they're just going to act apprehensive and possibly react negatively. But if someone were to ask, I like what Furry-Sneezes said about confidence. I'd answer honestly and confidently. And if they react negatively, that's their problem. Does all that help?
Anyway, sorry for the long post. I have the tendency to ramble.